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We have given the utility function

$\displaystyle U(v(c)-k(l))$

where $\displaystyle u$ and $\displaystyle v$ are increasing and concave functions and $\displaystyle k$ is increasing and convex. Each consumer has the income $\tilde{w_i}$ and another exogenous income $\displaystyle e$. In addition, the government charges a linear income tax, denominated as $\displaystyle \tau$.

The question is: How will the consumption $\displaystyle c(w_i,e)$, the labor supply $\displaystyle l(w_i,e)$ and the production $\displaystyle \tilde{w_i}l(w_i,e)$ react to a small increase of either the exogenous income $\displaystyle e$ or the net wage $\displaystyle w_i=(1-\tau)\tilde{w_i}$?

We have to use the implicit differentiation to prove the result, which is the main thing that I can't understand. I would have tried using the Lagrange method. Can you help me here?

Intuitively, an increase of the income will result in the increase of the consumption, as you have more money available. Regarding the labor supply, however, it depends very much on the substitution and the income effect. If the substitution effect is larger, then the consumer will prefer to work less.

The effect on the production is unclear. If the labor supply decreases faster than the wages increase, then the production will decrease. Otherwise it will increase.

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